# Ermakov-Painlev\'e II Symmetry Reduction of a Korteweg Capillarity System

## TL;DR

This paper demonstrates how a nonlinear Schrödinger equation modeling a Korteweg capillarity system can be reduced to an Ermakov-Painlevé II equation, leading to new exact solutions via Bäcklund transformations and linking to Painlevé equations.

## Contribution

It introduces a novel symmetry reduction of a capillarity system to an Ermakov-Painlevé II equation and constructs explicit solutions using Bäcklund transformations and special functions.

## Key findings

- Reduction to Ermakov-Painlevé II equation established
- Exact solutions derived using Bäcklund transformations and special functions
- Connection between capillarity system and Painlevé XXXIV equation identified

## Abstract

A class of nonlinear Schr\"odinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlev\'e II equation which is linked, in turn, to the integrable Painlev\'e XXXIV equation. A nonlinear Schr\"odinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlev\'e II reduction valid for a multi-parameter class of free energy functions. Iterated application of a B\"acklund transformation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials or classical Airy functions. A Painlev\'e XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion.